Faculty Publications, Department of Mathematics

THE COMPOSITION OF OPERATOR-VALUED MEASURABLE FUNCTIONS IS MEASURABLE

Albert Badrikian et al. — Mon, 13 Dec 2010 13:16:09 PST

Given separable Frechet spaces, E, F , and G , let L(E, F), L(F, G), and L(E, G) denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of point-wise convergence and the topology of compact convergence. We will show that if (X, F) is any measurable space and both A: X → L(E, F) and B: X → L(F, G) are Borelian, then the operator composition BA: X → L(E, G) is also Borelian. Further, we will give several consequences of this result.

Parameterizing the growth-decline boundary for uncertain population projection models

Joan Lubben et al. — Mon, 13 Dec 2010 13:07:50 PST

We consider discrete time linear population models of the form n(t + 1) = An(t) where A is a population projection matrix or integral projection operator, and represents a structured population at time t. It is well known that the asymptotic growth or decay rate of n(t) is determined by the leading eigenvalue of A. In practice, population models have substantial parameter uncertainty, and it might be difficult to quantify the effect of this uncertainty on the leading eigenvalue. For a large class of matrices and integral operators A, we give sufficient conditions for an eigenvalue to be the leading eigenvalue. By preselecting the leading eigenvalue to be equal to 1, this allows us to easily identify, which combination of parameters, within the confines of their uncertainty, lead to asymptotic growth, and which lead to asymptotic decay. We then apply these results to the analysis of uncertainty in both a matrix model and an integral model for a population of thistles. We show these results can be generalized to any preselected leading eigenvalue.

A PROOF OF NEWTON'S POWER SUM FORMULAS

J. A. Eidswick — Fri, 02 Apr 2010 12:26:30 PDT

For a polynomial P(z) = α₀ + α₁z + ... + α_nzⁿ = α_n (z – z₁) (z – z₂) ... (z – z_n), the power sums S_m = ∑ ⁿ _k=1 z^m _k, m = 1, 2, ... , can be calculated from the formulas ...

THE DIFFERENTIABILITY OF a^x

J. A. Eidswick — Fri, 02 Apr 2010 12:10:53 PDT

A "from scratch" proof of the differentiability of a^x, a > 0, is avoided by essentially all modern-day authors. A slick and popular way of handling the problem is to define a^x as e^x log a its differentiability and other properties following from that of the functions e^x and log x. Unfortunately, the usual definitions of e^x and log x involve relatively sophisticated ideas (e.g., integration or power series). Furthermore, the student, having heard of e, the natural logarithm base, at an early stage of his development, is hardly enlightened when he is told that e is e¹. He would have a much better feeling for the "naturalness" of e if it were defined as that number a for which (a^x)' = a^x.

The purpose of this note is to provide a direct and relatively simple way of getting at the differentiability of a^x.

A THEOREM ON REPEATING DECIMALS

William G. Leavitt — Fri, 02 Apr 2010 12:02:13 PDT

It is well known that a real number is rational if and only if its decimal expansion is a repeating decimal. For example, 2/7 = .285714285714 . . . . Many students also know that if n/m is a rational number reduced to lowest terms (that is, n and m relatively prime), then the number of repeated digits (we call this the length of period) depends only on m. Thus all fractions with denominator 7 have length of period 6. A sharp-eyed student may also notice that when the period (that is, the repeating digits) for 2/7 is split into its two half-periods 285 and 714, then the sum 285 + 714 = 999 is a string of nines. A little experimentation makes it appear likely that this is always true for a fraction with the denominator 7, as well as for fractions with denominators 11, 13, or 17. A natural conjecture is that all primes with even length of period (note that many primes, such as 3 and 31, have odd length of period) will have a similar property. This conjecture is, in fact, true but it is unfortunately not a criterion for primeness, since many composite numbers (such as 77) also have the property. The relevant theorem appears not to be well known, although it was discovered many years ago. (L. E. Dickson [see 1, p. 163] attributes the result to E. Midy, Nantes, 1836). The proof of the theorem is simple and elegant, and since it also provides a nice example of the usefulness of the concept of the order of an element of a group, it deserves to be better known.

In the following, we will develop from the beginning the theory of repeating decimals. This is to provide the necessary machinery for the proof of Midy's theorem, as well as for completeness.

MODULES OVER COMMUTATIVE RINGS

William G. Leavitt — Fri, 02 Apr 2010 11:57:16 PDT

The following is another short proof of the fact that for a commutative ring with unit R, any finitely based R-module is "dimensional" in the sense that all of its bases have the same number of elements.

THEOREM. Let R be a commutative ring with unit. If M is a unitary R-module with a basis of n elements, then all bases of M contain exactly n elements.

A NOTE ON HAUSDORFF SEPARATION

Edwin Halfar — Fri, 02 Apr 2010 11:52:23 PDT

The examples usually given as instances of topological spaces that have T₁-separation but not T₂-separation (Hausdorff) also have the property that some compact subset is not closed. This with the classic result concerning closedness of compact subsets of a Hausdorff space suggests the question of the equivalence of Hausdorff separation and the condition that the class of compact subsets be a subclass of the class of the closed subsets of a given space. The following is a simple result of this type and may be of some use in an introductory course in point set topology.

THEOREM. If X is a space satisfying the first axiom of countability, then a necessary and sufficient condition that X be a Hausdorff space is that the class of compact subsets of X be a subclass of the class of closed subsets of X.

Only the sufficiency need be considered here.

On a Theorem of Hölder

Donald W. Miller — Fri, 02 Apr 2010 11:47:19 PDT

A well-known result, due to Hölder [1], is the following: The symmetric group S_n, has outer automorphisms if and only if n = 6. The classical proof of the existence of a class of outer automorphisms of S₆, as formulated by Burnside [2], rests in part on the theory of primitive groups and entails extensive computation. In this note we offer a direct method for constructing such automorphisms. The author is grateful to Professor R. H. Bruck for raising this problem and for subsequent helpful remarks.

THE PRACTICAL EVALUATION OF RESULTANTS

T. A. Pierce — Fri, 02 Apr 2010 11:39:29 PDT

The purpose of the present note is to give a practical method of evaluating the resultant of two equations. The method is particularly effective when the degree of one of the equations is high while that of the other is low. Use will be made of certain results in the theory of matrices.

PROFESSOR LEO MOSER -- REFLECTIONS OF A VISIT

Walter E. Mientka — Fri, 02 Apr 2010 11:37:04 PDT

Professor Leo Moser' was known throughout the Mathematical Community as a significant researcher and excellent lecturer. I first met Leo during the Summer Research Institute in the Theory of Numbers held at the University of Colorado in 1959. After talking with him and hearing his lectures during the Institute, I felt that arrangements would have to be made in the near future for a visit to Nebraska. During the academic year 1962-63 while Professor Moser was on a lecture tour for the MAA, I invited him to present two research lectures to the Nebraska Section on May 3 and 4, 1963. He responded: "Professor D. W. Western of Franklin and Marshall College is my booking agent and I will write him immediately and find out whether it would be possible to clear May 3rd and 4th for me and thus enable me to give the lectures in Nebraska." His generosity was revealed in a subsequent letter in which he asserted: "According to a letter just received from Professor D. W. Western, I am to lecture in Cleveland, Ohio on May 1st and 2nd and in St. Petersburg, Florida on May 6th and 7th. Assuming connections are not too bad I should be able to get to Nebraska in time. If I find that the connections are not easy then I can move the Cleveland date back by one week I imagine. My talks at Nebraska will be on Number Theory and have the general title "Some New Applications of Generating Series." As usual his lectures were delivered with vigor, humor, and clarity. Following his last lecture I invited him to my office in order to discuss some of his results, and during our conversation the subject of mathematical limericks was mentioned and he asked if I would like to record some of his and other's limericks. (I had previously received his permission to record his lectures.) The main purpose of this paper is to present a transcription of these limericks and other verse, recorded on May 4, 1963.

Superspace geometrical realization of the N-extended super Virasoro algebra and its dual

Carina Curto et al. — Fri, 04 Sep 2009 08:59:02 PDT

Abstract We derive properties of N-extended GR super Virasoro algebras. These include adding central extensions, identification of all primary fields and the action of the adjoint representation on its dual. The final result suggest identification with the spectrum of fields in supergravity theories and superstring/M-theory constructed from NSR N-extended supersymmetric GR Virasoro algebras.

[The version deposited with arXiv (February 2000) is also attached (below) as an additional file.]

Matrix Model Superpotentials and Calabi–Yau Spaces: An A-D-E Classification

Carina Curto — Wed, 26 Aug 2009 10:09:13 PDT

We use F. Ferrari’s methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. The connection between matrix models and N = 1 gauge theories can be seen as evidence for the Dijkgraaf–Vafa conjecture. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau’s with corresponding ADE singularities. Moreover, in the additional Ô, Â, Dˆ and Ê cases we find new singular geometries. These ‘hat’ geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two coordinate charts. To obtain these results we develop techniques for performing small resolutions and small blow-downs, including an algorithm for blowing down exceptional P¹’s. In particular, we conjecture that small resolutions for isolated Gorenstein threefold singularities can be obtained by deforming matrix factorizations for simple surface singularities – and prove this in the length 1 and length 2 cases.

Auroral source region: Plasma properties of the high-latitude plasma sheet

C. A. Kletzing et al. — Wed, 26 Aug 2009 07:27:12 PDT

Statistical results from a survey of 93 passes through the high-latitude extension of the plasma sheet of electron data from the Hydra instrument on the Polar spacecraft show that the values for electron density can range from 0.01 to 0.5 cm^–3 with an average value around 0.1 cm^–3 on the poleward side and 0.3 cm^–3 on the equatorward side. Electron mean energy is found to have an average value near 900 eV on the equatorward side and 400 eV on the poleward side but varies from 100 eV to 4 keV. These values for density and mean energy are similar to those reported for measurements made in the equatorial plasma sheet by several previous spacecraft. The character of the electron distributions has been compared with Maxwellian and κ-distributions with the result that the κ-distribution with κ ≤ 10 yields an acceptable fit to the data twice as often as a Maxwellian distribution. This is similar to results found in the equatorial plasma sheet for both electrons and ions. The variation of electron density and mean energy around their average values have been compared with several solar wind parameters which have been developed to correlate solar wind variation with magnetospheric activity level. Few of these parameters are found to provide significant correlation with high-latitude plasma sheet electron density or temperature with the notable exception of solar wind density and solar wind particle flux which correlate with plasma sheet density.

Valuations for Spike Train Prediction

Vladimir Itskov et al. — Wed, 26 Aug 2009 07:18:48 PDT

The ultimate product of an electrophysiology experiment is often a decision on which biological hypothesis or model best explains the observed data. We outline a paradigm designed for comparison of different models, which we refer to as spike train prediction. A key ingredient of this paradigm is a prediction quality valuation that estimates how close a predicted conditional intensity function is to an actual observed spike train. Although a valuation based on log likelihood (L) is most natural, it has various complications in this context. We propose that a quadratic valuation (Q) can be used as an alternative to L. Q shares some important theoretical properties with L, including consistency, and the two valuations perform similarly on simulated and experimental data. Moreover,Q is more robust than L, and optimization with Q can dramatically improve computational efficiency. We illustrate the utility of Q for comparing models of peer prediction, where it can be computed directly from crosscorrelograms. Although Q does not have a straightforward probabilistic interpretation, Q is essentially given by Euclidean distance.

Threefold Flops via Matrix Factorization

Carina Curto et al. — Wed, 26 Aug 2009 07:15:30 PDT

The structure of birational maps between algebraic varieties becomes increasingly complicated as the dimension of the varieties increases. There is no birational geometry to speak of in dimension one: if two complete algebraic curves are birationally isomorphic then they are biregularly isomorphic. In dimension two we encounter the phenomenon of the blowup of a point, and every birational isomorphism can be factored into a sequence of blowups and blowdowns. In dimension three, however, we first encounter birational maps which are biregular outside of a subvariety of codimension two (called the center of the birational map). When the center has a neighborhood with trivial canonical bundle, the birational map is called a flop. The focus of this paper will be the case of a three-dimensional simple flop, in which the center is an irreducible curve (necessarily a smooth rational curve). One of the motivations for studying this case is a theorem of Kawamata [17], which says that all birational maps between Calabi–Yau threefolds can be expressed as the composition of simple flops (in fact, of simple flops between nonsingular varieties).

A Simple Model of Cortical Dynamics Explains Variability and State Dependence of Sensory Responses in Urethane-Anesthetized Auditory Cortex

Carina Curto et al. — Wed, 26 Aug 2009 07:08:03 PDT

The responses of neocortical cells to sensory stimuli are variable and state dependent. It has been hypothesized that intrinsic cortical dynamics play an important role in trial-to-trial variability; the precise nature of this dependence, however, is poorly understood. We show here that in auditory cortex of urethane-anesthetized rats, population responses to click stimuli can be quantitatively predicted on a trial-by-trial basis by a simple dynamical system model estimated from spontaneous activity immediately preceding stimulus presentation. Changes in cortical state correspond consistently to changes in model dynamics, reflecting a nonlinear, self-exciting system in synchronized states and an approximately linear system in desynchronized states. We propose that the complex and state-dependent pattern of trial-to-trial variability can be explained by a simple principle: sensory responses are shaped by the same intrinsic dynamics that govern ongoing spontaneous activity.

Supplementary Text to accompany “Cell Groups Reveal Structure of Stimulus Space”

Carina Curto et al. — Tue, 25 Aug 2009 15:23:16 PDT

Here we present a brief exposition of some material from algebraic topology that we use in our methods. We include it for completeness, as it may not be familiar for many readers. In particular, we define simplicial complexes, simplicial homology groups, and state the theorem cited in the Results section. See [Bott and Tu, 1982, Ewald, 1996, Hatcher, 2002] for more details.

Cell Groups Reveal Structure of Stimulus Space

Carina Curto et al. — Tue, 25 Aug 2009 15:20:33 PDT

An important task of the brain is to represent the outside world. It is unclear how the brain may do this, however, as it can only rely on neural responses and has no independent access to external stimuli in order to ‘‘decode’’ what those responses mean. We investigate what can be learned about a space of stimuli using only the action potentials (spikes) of cells with stereotyped—but unknown—receptive fields. Using hippocampal place cells as a model system, we show that one can (1) extract global features of the environment and (2) construct an accurate representation of space, up to an overall scale factor, that can be used to track the animal’s position. Unlike previous approaches to reconstructing position from place cell activity, this information is derived without knowing place fields or any other functions relating neural responses to position. We find that simply knowing which groups of cells fire together reveals a surprising amount of structure in the underlying stimulus space; this may enable the brain to construct its own internal representations.

Matrix model superpotentials and ADE singularities

Carina Curto — Tue, 25 Aug 2009 15:16:10 PDT

We use F. Ferrari’s methods relating matrix models to Calabi–Yau spaces in order to explain much of Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi–Yau with corresponding ADE singularities. Moreover, in the additional Ô, Â, Dˆ and Ê cases we find new singular geometries. These “hat” geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two co-ordinate charts. To obtain these results we develop an algorithm for blowing down exceptional P¹, described in the appendix.

Parametric Solutions of Certain Diophantine Equations

T. A. Pierce — Thu, 04 Jun 2009 11:30:01 PDT

In this note parametric solutions of certain diophantine equations are given. The method of obtaining the solutions is derived from an equation involving the determinants of certain matrices. It will be recognized that the method is a generalization of the method of Euler and Lagrange which depends on forms which repeat under multiplication. The matrices used in this paper must be such that their forms are retained under matric multiplication and addition. When integer values are assigned to the parameters of our solutions we obtain integer solutions of the particular equation under consideration; however not all integer solutions are necessarily furnished this way.